Finite Group Databasebeta

← #28364714

Subgroups — #28364714

Order 13952.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295575C211yesmaximal, characteristic
2#20295009C212yesmaximal, complemented
3#20294824C212yesmaximal, complemented
4#20294823C211yesmaximal, characteristic, complemented
5#355×C109C211yesmaximal, characteristic, complemented
6C2×#13182030C2211yescharacteristic
7C2×#13182028C411yescharacteristic
8M16×D109—22no—
9#16812697C2211yescharacteristic
10#16812695C411yescharacteristic
11#16812687C411yescharacteristic
12#16812672—22no—
13#16812659—22no—
14C42×D109C411yescharacteristic
15#16812618C2212yes—
16#16812615—22no—
17#124×C109C2211yescharacteristic
18#101×C109C2212yes—
19#13182030—41no—
20#13182030—41no—
21#13182028—41no—
22#13182028—41no—
23C2×Dic218C2×C411yescharacteristic
24C2×D436C2×C411yescharacteristic
25C2×D436D412yes—
26C2×C22⋊D109—22no—
27C2×C22⋊D109—21no—
28C22×Dic109—21no—
29C2×C4×D109—21no—
30C2×C4×D109C2×C411yescharacteristic
31C2×C4×D109D412yes—
32#13182013—22no—
33#13182012—42no—
34C8×D109—42no—
35#13182008—22no—
36#13182006—22no—
37C4×Dic109—21no—
38C4×Dic109C2×C411yescharacteristic
39C2×Q8×C109C2311yescharacteristic
40C2×D4×C109C2×C411yescharacteristic
41M16×C109—22no—
42C22⋊C4×C109—22no—
43C42×C109C2×C411yescharacteristic
44C23×C109—21no—
45C22×D109C22⋊C412yes—
46C22×D109C22⋊C411yescharacteristic
47Q8×C109—41no—
48D4×C109—41no—
49C2×C4×C109—21no—
50C2×C4×C109C22×C411yescharacteristic
51C2×C4×C109C2×D412yes—
52C22⋊D109—44no—
53C22⋊D109—42no—
54C2×Dic109—21no—
55C2×Dic109—21no—
56C2×Dic109—41no—
57C2×Dic109—41no—
58C2×Dic109C22⋊C411yescharacteristic
59D436—41no—
60D436—42no—
61C4×D109—22no—
62C4×D109—22no—
63C4×D109—21no—
64C4×D109—21no—
65C4×D109—42no—
66Dic218—41no—
67C8×C109—42no—
68#11724784—42no—
69C22×C109—42no—
70C22×C109—41no—
71C22×C109C2×C22⋊C411yescharacteristic
72C2×D109—21no—
73C2×D109—22no—
74C2×D109—41no—
75C2×D109—42no—
76C4×C109—22no—
77C4×C109—21no—
78C4×C109—41no—
79Dic109—22no—
80Dic109—41no—
81Dic109#9911yescharacteristic
82Dic109#9911yescharacteristic
83C2×C109—21no—
84C2×C109—81no—
85C2×C109C2×#9911yescharacteristic
86D109—41no—
87D109—42no—
88#2132—1091nomaximal, Sylow
89C109#213211yescharacteristic, complemented, Sylow
90#532—1091no—
91#413—1092no—
92#356—1092no—
93#355—1091no—
94#355D10911yescharacteristic, complemented
95C2×#41—1091no—
96C2×#41—1091no—
97C2×M16—2182no—
98C4⋊Q8—1091no—
99#127—1091no—
100#124—1091no—
101#124C2×D10911yescharacteristic
102#121—2182no—
103C4×D4—2182no—
104C2×C42—1091no—
105#101—1092no—
106#101C2×D10912yes—
107#100—2182no—
108#41—4361no—
109#41—4361no—
110#41—4361no—
111#41—4361no—
112C2×Q8—1091no—
113C2×Q8C22×D10911yescharacteristic
114C2×D4—1091no—
115C2×D4—1092no—
116C2×D4—2182no—
117C2×D4—2181no—
118C2×D4C4×D10911yescharacteristic
119C22×C4—1091no—
120C22×C4—1092no—
121C22×C4—2181no—
122C22×C4—2181no—
123M16—22no—
124M16—4362no—
125M16—2182no—
126C2×C8—4362no—
127C4⋊C4—2182no—
128C22⋊C4—22no—
129C22⋊C4—2182no—
130C42—1091no—
131C42—2181no—
132C42C4×D10911yescharacteristic
133C23—21no—
134C23—1092no—
135C23—1091no—
136Q8—41no—
137Q8—4361no—
138D4—41no—
139D4—4362no—
140D4—4361no—
141D4—4362no—
142D4—4364no—
143C2×C4—21no—
144C2×C4—1091no—
145C2×C4—4362no—
146C2×C4—4361no—
147C2×C4—4361no—
148C2×C4—2182no—
149C2×C4—2182no—
150C2×C4—2181no—
151C2×C4—2181no—
152C2×C4—2181no—
153C2×C4—2181no—
154C2×C4C2×C4×D10911yescharacteristic
155C2×C4D4×D10912yes—
156C8—42no—
157C8—4362no—
158C22—42no—
159C22—41no—
160C22—4361no—
161C22—4362no—
162C22—2181no—
163C22—2182no—
164C22C22⋊C4×D10911yescharacteristic
165C4—22no—
166C4—21no—
167C4—41no—
168C4—1091no—
169C4—1091no—
170C4—4361no—
171C4—2182no—
172C2—21no—
173C2—81no—
174C2—4361no—
175C2—4362no—
176C2#99×D10911yescharacteristic
subgroupG/H
centerC2#99×D109
commutatorC2×C4×C109C22×C4
FrattiniC2×Q8C22×D109
Fitting#355×C109C2
socleC2×C109C2×#99
prime radical#20295575C2
cosocleC2×Q8×C109C23
hypercenter#355D109
hypocenterC109#2132
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20295009C2semidirect
3. #20294824C2semidirect
4. #20294823C2semidirect
5. #355×C109C2semidirect
89. C109#2132semidirect
94. #355D109semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202955752
#202950092
#202948242
#202948232
#355×C1092
#2132109